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User blog:B1mb0w/new Alpha Function
'(new) Alpha Function' The Alpha Function has one parameter: \(\alpha®\) where r is any real number. It is derived from the Strong D Function with a variable number of input parameters. This blog replaces the old description of the Alpha Function. 'What is the Alpha Function' My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(1)\) for example can be used to reference the number 2. Therefore 1 is the Alpha Index for the number 2. Alpha needs to reference big numbers very quickly to be useful, therefore it uses the Strong D Function for this purpose. Alpha should also be strictly hierarchical and for every number a > b: a must reference larger numbers so that \(\alpha(a) >> \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and offers some finesse to locate large big numbers. The Alpha Function is defined recursively based on a series of binary decisions. The logic can be explained by referring to the following notation. 'Some Notation' Following notation helps to explain the behaviour of Strong D Functions and the logic of the Alpha Function. \(D(m_{x}) = D(m_1,m_2,...,m_x)\) \(D(m_{x},n_{y}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\) \(D(1,0_{y}) = D(D(1_{y})_{y})\) \(D(m,n_{y}) = D(m-1,D(m,n_{y-1},n_y-1)_{y})\) \(D(m_{x},n,0_{y}) = D(m_1-1,D(m_{x},n-1,n_{y})_{x+y})\) when x>0 and which is equal to \(= D(m_{x},n-1,n_{y-1},n_y+1)\) 'Alpha Function Logic' The Alpha Function is defined using the following logic. \(\alpha® = D(D(m_{x})_{x})\) where \(2^{x-1} <= r < 2^x\) and \(\alpha(2^x) = D(1, 0_{x})\) The values of \(m_{x}\) are calculated based on the value of r but only legal values can be used which follow these restrictions: *duplicate values should be avoided *out of sequence results must be avoided The second constraint is important to force \(\alpha(a) >> \alpha(b)\) whenever a > b. Additional logic that is used is derived from the following rules: Maximum Value Rule: M1 \(D(1,0_{x}) = D(D(1_{x})_{x})\) therefore \(D(m_{x}) < D(D(1_{x})_{x})\) \(<= D(D(1_{x})_{x})-1\) or alternatively \(<= D(D(1_{x})_{x-1},D(1_{x-1},0))\) Rule M1a and \(m_1 >= 1\) Rule M1b Maximum Value Rule: M2 \(D(m_{x},n+1,0_{y}) = D(m_{x},n,n_{y-1}+1,n_y+2)\) therefore \(D(m_{x},n,p_{y}) < D(m_{x},n,n_{y-1}+1,n_y+2)\) \(<= D(m_{x},n,n_{y-1}+1,n_y+2)-1\) or alternatively \(<= D(m_{x},n,n_{y}+1)\) Rule M2a or \(p_{y} <= n_{y}+1\) Rule M2b Maximum Value Rule: M3 It can also be shown that the only legal values for D functions in the form: \(D(m_{x})\) are when \(m_i <= m_{i-1}+1\) for all \(1 <= i <= x\) Maximum Value Rule: M4 The final rule is used for D functions of the form: \(D(n+1,0_{y}) = D(n,D(n,n_{y}+1)_{y}\) therefore \(D(n,p_{y}) < D(n,D(n,n_{y}+1)_{y})\) \(<= D(n,D(n,n_{y}+1)_{y}-1)\) or alternatively \(<= D(n,D(n,n_{y}+1)_{y-1},D(n,n_{y-1}+1,n))\) Rule M4a and \(p_i <= D(n,n_{y}+1)\) for all \(1 <= i <= y\) Rule M4b and \(p_y <= D(n,n_{y-1}+1,n)\) Rule M4c 'Some Calculations' \(\alpha(0) = D() = 0\) \(\alpha(1) = D(1) = 2\) \(\alpha(2) = D(1,0) = 3\) \(\alpha(4) = D(1,0,0)\) 'Program Code and Description' Version 1 of the program code for the Alpha Function will be available here (work in progress). The code is not complete and various errors will be corrected in Version 2 (work in progress). 'Comments and Questions' Look forward to comments and questions. I am learning heaps by writing these blogs and correcting all the mistakes the community finds in them ! Cheers B1mb0w. 'References' The Alpha Function *''The (old) Alpha Function'' *Strong D Function **Deeply Nested Ackermann Function ***Modified Ackermann Function **Strong D Function Calculations **Mapping D(l,0,1) to epsilon nought **To Be Updated Category:Blog posts